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On the structure of the conformal Gaussian curvature equation on \(R^ 2\). II. (English) Zbl 0753.35032
This paper is a sequel to a previous work of the authors [part I, Duke Math. J. 62, No. 3, 721-737 (1991; Zbl 0733.35037)], where they study the problem of finding a metric on \(\mathbb{R}^ 2\) conformal with the Euclidean one and having a prescribed scalar curvature \(K(x)\). The authors are able to obtain incredibly accurate classification results provided that \(K\) satisfies a certain power-like decaying at infinity. Then one can find a whole family of conformal metrics with curvature \(K\). This family is bijectively determined by a parameter \(\alpha\) which describes the asymptotic behaviour of the metric at infinity. The method of proof is a clever and delicate adaptation of Perron’s method of sub- and super solutions.

MSC:
35J60 Nonlinear elliptic equations
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
35B20 Perturbations in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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References:
[1] Cheng, K.-S., Ni, W.-M.: On the structure of the conformal Gaussian curvature equation onR 2. I. Duke Math. J.62, 721-737 (1991) · Zbl 0733.35037 · doi:10.1215/S0012-7094-91-06231-9
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